Analytic solutions of a generalized complex multi-dimensional system with fractional order

Dumitru Baleanu; Rabha W. Ibrahim

doi:10.1515/dema-2024-0029

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Analytic solutions of a generalized complex multi-dimensional system with fractional order

Dumitru Baleanu and Rabha W. Ibrahim

Published/Copyright: January 27, 2025

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From the journal Demonstratio Mathematica Volume 58 Issue 1

Abstract

The Duhamel principle is a mathematical principle that allows us to solve linear partial differential equations. This system is generalized by the concept of the k -symbol fractional calculus. We demonstrate that in specific functional domains, the suggested system produces a global solution. Convergence toward the stable point is investigated. Some special cases are illustrated including the rotated Koebe function, which is used for creating the optimal solutions, where these functions are convex univalent in the open unit disk.

Keywords: fractional differential equations; fractional differential operator; K-symbol calculus; fractional calculus; analytic functions; the Cauchy-Schwartz inequality

MSC 2010: 30C45; 30C55

1 Introduction

The Duhamel principle is useful in many areas of science and engineering, including physics, acoustics, and fluid mechanics. It provides a powerful tool for solving linear partial differential equations in a general and elegant way [1]. K -symbol fractional calculus is the generalization of the gamma function to include the parameter K [2]. This generalization leads to extended the well-known fractional differential and integral operators. This extension covers the fractional operators of a complex variable K -symbol fractional calculus is a powerful tool for analyzing and modeling complex systems that exhibit noninteger order dynamics. It provides a systematic way of defining and computing fractional derivatives and integrals of arbitrary order, which enables us to study a wide range of phenomena in various fields of science and engineering [3–5].

The existence and uniqueness of global solutions depend on the specific problem at hand. In general, the existence and uniqueness of global solutions to a system of equations or a differential equation can be proven under certain conditions [6,7]. Similarly, for partial differential equations, the existence and uniqueness of global solutions can be established under certain conditions, such as the partial differential equation (PDE) being well-posed, meaning that it has a unique solution that depends continuously on the initial or boundary conditions. In general, proving the existence and uniqueness of global solutions requires a deep understanding of the specific problem at hand and often involves advanced mathematical techniques. In some cases, the existence and uniqueness of global solutions cannot be proven, and one must resort to numerical methods to approximate and analytic solutions [8–10].

The existence and uniqueness of a strong global-in-time solution inside suitable functional domains. Concepts of the existence proof include explicit energy predictions, transition to the limit, estimates on a series of approximations, and long-term pattern that relies on the boundary condition and the size at which fluidity disappears (solid fluid with fluidity φ = 0 ) is presented for the 2D-system [11] (for recent work, see [12])

∂ τ υ ( τ , y ) = Λ 1 ( υ , ς ) ∂ τ ς ( τ , y ) = Λ 2 ( υ , ς ) ,

where υ is the velocity and ς is the shear stress for τ ∈ [ 0 , ⊤ ] and y ∈ [ 0 , 1 ] . The 3D system is investigated in [13] (recent work with details) by adding the fluidity φ , as a variable in the time-space ( τ , y ), as follows:

∂ τ υ ( τ , y ) = Λ 1 ( υ , ς , φ ) ∂ τ ς ( τ , y ) = Λ 2 ( υ , ς , φ ) ∂ τ φ ( τ , y ) = Λ 3 ( υ , ς , φ ) .

In this work, we have made an extension to the space variable to be a complex variable in a complex domain (the unit disk). Moreover, a generalization of the derivative is indicated by using the K -symbol Riemann-Liouville fractional calculus [14–16]. We propose a set of conditions for the existence and uniqueness of the multicoefficients 3D-K-symbol fractional system. We demonstrate how the suggested system, in the complex domains, implies a global solution. It is indicated that this is a strong and longtime behavior of the outcomes. We use an approach that is based on the fractional Duhamel concept and analytic techniques from fixed point theory. Special cases of the optimal solutions are introduced by using the rotated Koebe function. These analytic functions are convex univalent in the open unit disk.

2 K -symbol method

To modify the well-known fractional calculus based on the k -gamma function, the idea of k -symbol calculus is introduced, for the first time, in [2] (Table 1)

Γ k ( ξ ) = ∫ 0 ∞ exp − ρ k k ρ ξ − 1 d ρ , ℜ ( ξ ) > 0 , k ∈ R + .

Table 1

Plot of K -symbol gamma function, for K = 1 , 2 , 3 , 4 respectively, where the relation between the normal gamma and the K-symbol gamma is Γ k ( ξ ) = k ξ k ‒ 1 Γ ξ k . The plot is given by Mathematica 13.3

k	Γ k ( ξ )	Plot
1	Γ 1 ( ξ ) = Γ ( ξ )
2	Γ 2 ( ξ ) = 2 ξ ⁄ 2 ‒ 1 Γ ( ξ ⁄ 2 ) , ℜ ( ξ ) > 0
3	Γ 3 ( ξ ) = 3 ξ ⁄ 3 ‒ 1 Γ ( ξ ⁄ 3 ) , ℜ ( ξ ) > 0
4	Γ 4 ( ξ ) = 2 ξ ⁄ 2 ‒ 2 Γ ( ξ ⁄ 4 ) , ℜ ( ξ ) > 0

Definition 2.1

For a nonnegative real number k , the generalized k -gamma function is structured by the integral formula:

(2.1) Γ k ( ξ ) = lim n → ∞ n ! k n ( n k ) ξ k − 1 ( ξ ) n , k ,

where

( ξ ) n , k ≔ ξ ( ξ + k ) ( ξ + 2 k ) … ( ξ + ( n − 1 ) k )

and

( ξ ) n , k = Γ k ( ξ + n k ) Γ k ( ξ ) .

Obviously, Γ k ( ξ ) → Γ ( ξ ) whenever k → 1 , and

Γ k ( ξ + k ) = ξ Γ k ( ξ ) , Γ k ( k ) = 1 , Γ k ( ξ ) = k ξ k − 1 Γ ξ k .

By employing Γ k ( ξ ) , the Riemann-Liouville operators are generalized in [14,15], as follows:

Definition 2.2

The k -symbol Riemann-Liouville fractional integral admits the following formula:

L a , k ν h ( ξ ) = 1 k Γ k ( ν ) ∫ a ξ ( ξ − z ) ν ⁄ k − 1 h ( z ) d z , ν > 0 .

Perceive that, for h ( ξ ) = 1 , this implies that

L a , k ν h ( ξ ) = ( ξ − a ) ν k Γ k ( ν + k ) , ν ∈ ( 0 , 1 ] .

When a = 0 , a specific case is recorded that

L k ν h ( ξ ) = 1 k Γ k ( ν ) ∫ 0 ξ ( ξ − z ) ν ⁄ k − 1 h ( z ) d z .

The k -symbol Riemann-Liouville fractional derivative is known by

Δ k ν h ( ξ ) = d d ξ ( L k 1 − ν h ( ξ ) ) .

The Caputo derivative with sign k may be calculated using the following equation:

(2.2) k C D ν h ( ξ ) = k v L k v k − ν h ( v ) ( ξ ) = k v − 1 Γ k ( v k − ν ) ∫ 0 ξ h ( v ) ( z ) ( ξ − z ) ν ⁄ k − v + 1 d z , ν ∈ ( ( v − 1 ) k , v k ] d v d ξ v h ( ξ ) , v k = ν .

Note that whenever k = 1 , the usual Caputo fractional operator is recognized. In addition, the function of the k -Riemann-Liouville singular kernel is formulated by

k ν , k ( ξ ) = ξ ν ⁄ k − 1 k Γ k ( ν ) , ν ∈ ( 0 , 1 ) , k > 0 .

Thus, in view of Hadamard convolution product, the integral operator becomes

L k ν h ( ξ ) = k ν , k ( ξ ) ⁎ h ( ξ ) .

For example, let h ( ξ ) = 1 , then

Δ k ν h ( ξ ) = ξ 1 − ν k − 1 Γ k ( 1 − ν ) .

In general, if h ( ξ ) = ξ m , then

Δ k ν h ( ξ ) = Γ k ( k ( m + 1 ) ) Γ k ( 1 − ν + k m ) ξ 1 − ν k + m − 1 .

It is clear that when k = 1 , the integral reduces to the normal definition [17]

Δ 1 ν h ( ξ ) = Γ ( m + 1 ) Γ ( 1 − ν + m ) ξ m − ν .

3 Results

Laminar-Couette flow (LCF) is a type of fluid flow that occurs between two parallel plates with one plate moving and the other plate being stationary. This flow is named after Maurice Marie Alfred Couette, who first studied it in 1890. In LCF, the fluid particles move in parallel layers, with each layer sliding over the adjacent layer. The layer of fluid in direct contact with the moving plate is dragged along with it, creating a velocity gradient across the fluid, which increases as the distance from the moving plate increases. The flow is characterized by the Reynolds number (Re), which is a dimensionless parameter that describes the ratio of inertial forces to viscous forces in the fluid. At LCF, the Reynolds number is low, indicating that viscous forces are dominant over inertial forces. LCF has important applications in various fields, including engineering, physics, and biology. As an example, LCF is frequently utilized in the design of lubrication systems, such as those found in automobile or aircraft engines. It is also used to study the behavior of blood flow in blood vessels and the motion of cilia in the respiratory system.

The 3D scenario we have suggested in this section corresponds physically to the consideration of an LCF. This kind of flow accurately simulates the flows in shear radiometers. Consider the three functions in ( τ , ζ ) as follows: the velocity υ , the shear stress ς , and the fluidity φ which are systematized as variable functions in ζ ∈ O ¯ ≔ { ζ ∈ C , ∣ ζ ∣ ≤ 1 } . Moreover, we suggest them as a function of time τ ≥ 0 . Thus, we obtain the following 3D- k -symbol fractional system:

(3.1a) α Δ k , τ ν υ ( τ , ζ ) = β υ ζ ζ + ς ζ ,

(3.1b) γ Δ k , τ ν ς ( τ , ζ ) = δ υ ζ − φ ς + δ ε ,

(3.1c) Δ k , τ ν φ ( τ , ζ ) = ( − 1 + ϑ ∣ ς ∣ ) φ 2 − μ φ 3 ,

where τ ∈ [ 0 , ⊤ ] , ⊤ < ∞ , ν ∈ ( 0 , 1 ) , α , β , γ , δ indicate the nonzero density, viscosity, relaxation time, and the elastic modulus, respectively, while ε is a nonnegative constant and ϑ and μ are the nonzero evolution of the fluidity φ .

Assume that the υ , ς , and φ are analytic over O ¯ such that ∣ φ ∣ ≤ 1 . System (3.1), which has three equations and seven dimensionless coefficients, is categorized as a fully linked system. Positive and consistent throughout time are all of the aforementioned coefficients. Equation (3.1a) is the momentum conservation equation for υ . Equation (3.1b) controls how the shear stress ς changes over time. Equation (3.1c) follows a pattern that several writers have proposed for evolution equations. Equations (3.1a) and (3.1b) are of a classical character; however, the final equation might vary from model to model.

Suppose that System (3.1) subjected to the initial condition ( υ 0 , ς 0 , φ 0 ) and the homogeneous boundary conditions (HBC) υ ( τ , 0 ) = 0 and υ ( τ , 1 ) = 0 . A fundamental physical value can be expressed as the outcome of the fundamental physical dimensions of distance lengthwise, mass, power, heat, and time, each raised to a rational power ( ℓ , Ω , Φ , Σ , and ⊤ ). These fundamental physical dimensions can be used to explain other physical quantities. For instance, velocity, stress, and fluidity all have the dimension of distance lengthwise per unit of time. These physically dependent values typically require constant coefficients. These coefficients typically have nonnegative values since they are consistent across time.

3.1 Solvability of system (3.1)

In this part, we aim to discuss the solvability of system (3.1) subjected to the initial condition ( υ 0 , ς 0 , φ 0 ) and HBC υ ( τ , 0 ) = 0 and υ ( τ , 1 ) = 0 for all τ ∈ [ 0 , ⊤ ] .

Theorem 3.1

Consider ( υ 0 , ς 0 , φ 0 ) ∈ H 1 ( O ¯ ) 3 with ℜ ( φ 0 ) ≥ 0 . If system (3.1) satisfies the inequality

⊤ ν ⁄ k ( 1 + μ ) k Γ k ( ν + 1 ) < 1 , ⊤ < ∞ , ν ∈ ( 0 , 1 ] ;

then it admits a unique global outcome ( υ , ς , φ ) under HBC υ ( τ , 0 ) = 0 and υ ( τ , 1 ) = 0 , where

(3.2) ( υ , ς , φ ) ∈ ( C ( [ 0 , ⊤ ] ; H 1 ) ∩ L 2 ( [ 0 , ⊤ ] ; H 2 ) × C ( [ 0 , ⊤ ] ; H 1 ) × C ( [ 0 , ⊤ ] ; H 1 ) )

and ℜ ( φ ) ≥ 0 for all ζ ∈ O ¯ and τ ∈ [ 0 , ⊤ ] . Moreover,

(3.3) ( Δ k , τ ν υ , Δ k , τ ν ς , Δ k , τ ν φ ) ∈ ( L 2 ( [ 0 , ⊤ ] ; L 2 ) × C ( [ 0 , ⊤ ] ; L 2 ) × C ( [ 0 , ⊤ ] ; L 2 ) ) .

Proof

There are eight steps in our proof. The first five steps arrive to the formal solution, and Step 6 describes the approximate solution’s step-by-step flow. Step 7 proves the convergence of this sequence, establishing the existence of a solution to System (3.1). Step 8 deals with distinctiveness.

Step 1. Non-negativity

Since ℜ ( φ 0 ) ≥ 0 and the analytic function φ ( ⋅ , ζ ) satisfies ∣ φ ∣ ≤ 1 over O ¯ , then by the maximum modulus principle for fluid [18],

φ ( r ) ≥ ∣ φ ( ζ ) ∣ ≥ 0 , ∀ ∣ ζ ∣ ≤ r ,

this gives that ℜ ( φ ( ⋅ , ζ ) ) ≥ 0 .

Step 2. Limitations of the outcome. Now, by multiplying by υ and integrating the first equation on υ over the domain in System (3.1) yields

(3.4) α ν ⁄ k + 1 Δ k , τ ν ‖ υ ( τ , ⋅ ) ‖ L 2 2 + β ‖ υ ζ ( τ , ⋅ ) ‖ L 2 2 = ∫ O ¯ ( ς ζ υ ) ( τ , ⋅ ) .

In the same manner for the second evolution equation in System (3.1), multiplying by ς and integrating over the domain, which implies

(3.5) γ ν ⁄ k + 1 Δ k , τ ν ‖ ς ( τ , ⋅ ) ‖ L 2 2 + ‖ φ ς ( τ , ⋅ ) ‖ L 2 2 = δ ∫ O ¯ ( ς υ ζ ς ) ( τ , ⋅ ) + δ ε ς ^ ,

where

φ ^ ( τ ) = ∫ O ¯ φ ( τ , ζ ) d ζ .

Combining estimates (3.4) and (3.5) and using the generalization of integration by part ([19]), where υ vanishes on the boundary, to obtain

(3.6) 1 ν ⁄ k + 1 Δ k , τ ν [ δ α ‖ υ ( τ , ⋅ ) ‖ L 2 2 + γ ‖ ς ( τ , ⋅ ) ‖ L 2 2 ] + ‖ φ ς ( τ , ⋅ ) ‖ L 2 2 + δ β ‖ υ ζ ( τ , ⋅ ) ‖ L 2 2 = δ ε ς ^ ( τ ) .

Integration of the third equation in System (3.1) over O ¯ implies

(3.7) Δ k , τ ν ‖ φ ( τ , ⋅ ) ‖ L 1 + ‖ φ ( τ , ⋅ ) ‖ L 2 2 + μ ‖ φ ( τ , ⋅ ) ‖ L 3 3 = ϑ ∫ O ¯ ( ∣ ς ∣ φ 2 ) ( τ , ⋅ ) .

The Young integral inequality ([20], Theorem 0.3.1) brings

ϑ ∣ ς ∣ φ 2 = μ ∣ φ ∣ 3 2 ϑ μ ∣ ς ∣ ∣ φ ∣ 1 2 ≤ μ 2 ∣ φ ∣ 3 + ϑ 2 2 μ ∣ φ ∣ ς 2 ,

which leads to the inequality

(3.8) Δ k , τ ν ‖ φ ( τ , ⋅ ) ‖ L 1 + ‖ φ ( τ , ⋅ ) ‖ L 2 2 + μ 2 ‖ φ ( τ , ⋅ ) ‖ L 3 3 ≤ ϑ 2 2 μ ‖ φ ς ( τ , ⋅ ) ‖ L 2 2 .

By adding up (3.6) and (3.8), we obtain

(3.9) 1 ν ⁄ k + 1 Δ k , τ ν δ α ‖ υ ( τ , ⋅ ) ‖ L 2 2 + γ ‖ ς ( τ , ⋅ ) ‖ L 2 2 + 2 μ ϑ 2 ‖ φ ( τ , ⋅ ) ‖ L 1 + 1 2 ‖ φ ς ( τ , ⋅ ) ‖ L 2 2 + δ β ‖ υ ζ ( τ , ⋅ ) ‖ L 2 2 ≤ C ε ‖ ς ( τ , ⋅ ) ‖ L 2 ,

where C > 0 depends on the constant connections α , β , γ , δ , μ , and ϑ . By using the inequality

‖ ς ( τ , ⋅ ) ‖ L 2 ≤ ‖ ς ( τ , ⋅ ) ‖ L 2 2 + 1 2 ,

then the generalized fractional Gronwall lemma on equation (3.9) yields [21]

(3.10) sup τ ∈ [ 0 , ⊤ ] [ ‖ υ ( τ , ⋅ ) ‖ L 2 2 + ‖ ς ( τ , ⋅ ) ‖ L 2 2 + ‖ φ ( τ , ⋅ ) ‖ L 1 ] + ⊤ ν ⁄ k − 1 k Γ k ( ν ) ∫ 0 ⊤ ( ‖ φ ς ( τ , ⋅ ) ‖ L 2 2 + ‖ υ ζ ( τ , ⋅ ) ‖ L 2 2 ) d τ ≤ C ˜ ,

where C ˜ > 0 depends on seven connection constants δ , ⊤ , υ 0 , ς 0 , φ 0 , ν , β , γ , μ , ϑ , α , and ε , such that when ε = 0 , in (3.9), then C ˜ is a constant free of ⊤ . Therefore, the uniformly boundedness in time is obtained.

Step 3. Further new integral functions. Let p be defined as follows:

p ( τ , ζ ) = ∫ 0 ζ ( ς ( τ , z ) − ς ^ ( τ ) ) d z ,

owning the Dirichlet boundary conditions, which satisfies

∂ 2 p ∂ ζ 2 = ∂ ς ∂ ζ .

The first and the second equations in System (3.1) impose

Δ k , τ ν υ = β α ∂ 2 ∂ ζ 2 υ + p β

and

γ Δ k , τ ν p = − ∫ 0 ζ ( φ ς ( τ , z ) − φ ς ^ ( τ ) ) d z + δ υ .

Formulate the functional Ψ as follows:

(3.11) Ψ = υ + 1 β ∫ 0 ζ ( ς − ς ^ ) d z = υ + p β .

A k -symbol fractional derivative yields

(3.12) Δ k , τ ν Ψ = Δ k , τ ν υ + 1 β Δ k , τ ν p = β α ∂ 2 ∂ ζ 2 ( υ + p β ) + 1 γ β − ∫ 0 ζ ( φ ς ( τ , z ) − φ ς ^ ( τ ) ) d z + δ υ = β α ∂ 2 ∂ ζ 2 Ψ − 1 γ β ∫ 0 ζ ( φ ς ( τ , z ) − φ ς ^ ( τ ) ) d z + δ β γ υ .

By multiplying equation (3.12) by ∂ 2 ∂ ζ 2 Ψ and integrating over O ¯ , we obtain

1 ν ⁄ k + 1 Δ k , τ ν ∂ Ψ ∂ ζ ( τ , ⋅ ) L 2 2 + β 2 α ∂ 2 Ψ ∂ ζ 2 ( τ , ⋅ ) L 2 2 ≤ C ‖ ( φ ς ) ( τ , ⋅ ) ‖ L 1 ∫ O ¯ ∂ 2 Ψ ∂ ζ 2 ( τ , ⋅ ) + ∫ O ¯ υ ∂ 2 Ψ ∂ ζ 2 ( τ , ⋅ ) .

By combining the Young integral inequality and the Cauchy-Schwarz inequality yields [22]

(3.13) Δ k , τ ν ∂ Ψ ∂ ζ ( τ , ⋅ ) L 2 2 + ∂ 2 Ψ ∂ ζ 2 ( τ , ⋅ ) L 2 2 ≤ C α , β , ν ⁄ k ( ‖ φ ( τ , ⋅ ) ‖ L 1 ‖ φ ς ‖ L 2 2 + ‖ υ ( τ , ⋅ ) ‖ L 2 2 ) .

But

∂ Ψ ∂ ζ ∣ τ = 0 = ∂ υ 0 ∂ ζ + 1 β ( ς 0 − ς ^ 0 ) ∈ L 2 ( O ¯ ) ;

then from (3.13), we obtain

Ψ ∈ L ∞ ( [ 0 , ⊤ ] , H 1 ) ∩ L 2 ( [ 0 , ⊤ ] , H 2 ) .

As a consequence, we obtain the result of this step.

(3.14) υ ∈ L ∞ ( [ 0 , ⊤ ] , H 1 ) ∩ L 2 ( [ 0 , ⊤ ] , H 2 ) .

Step 4. Using L ∞ -limits. From the formula of Ψ and ς ^ , we rearrange the second equation in System (3.1), as follows:

Ψ Δ k , τ ν ς = δ ∂ Ψ ∂ ζ − φ + δ β ς + δ β ς ^ + δ ε .

A multiplication by ϕ yields

γ Δ k , τ ν ∣ ς ∣ 2 + ∣ φ ∣ + δ β ∣ ς ∣ 2 ≤ C ∣ ς ∣ ∂ Ψ ∂ ζ + ∣ ς ∣ ‖ ς ‖ L 2 + ε ∣ ς ∣ .

Again, Young inequality imposes

(3.15) λ Δ k , τ ν ∣ ς ∣ 2 + ∣ φ ∣ + δ β ∣ ς ∣ 2 ≤ C ∂ Ψ ∂ ζ 2 + ‖ ς ‖ L 2 + ε .

But ς 0 ∈ H 1 (Step 2) and ∂ Ψ ∂ ζ ∈ L 2 ( [ 0 , ⊤ ] , L ∞ ) (relation (3.14)), then the modified Gronwall inequality yields

(3.16) ‖ ς ( τ , ⋅ ) ‖ L ∞ ≤ C ˜ ,

where C ˜ is given in (3.10), which leads to ς ∈ L ∞ ( [ 0 , ⊤ ] , L ∞ ) .

Next, we show that φ ∈ L ∞ ( [ 0 , ⊤ ] , L ∞ ) . By the fractional Duhamel principle [23], equation (3.1c) implies

Δ k , τ ν φ ( τ , ζ ) = ( − φ − ν φ 2 ) φ + ϑ ∣ ς ∣ φ 2 .

Consider the solution ℵ of the next equation

(3.17) Δ k , τ ν ℵ + ( ℵ + μ ℵ 2 ) ℵ = 0

with the initial condition

I 1 − ν ℵ ∣ τ = 0 = ℏ ( 0 ) , where ℏ ≔ ϑ ∣ ς ∣ φ 2 , ∣ ℵ ∣ ≤ 1 .

Thus,

φ ( τ ) = ∫ 0 τ ℵ ( s ) d s

is a solution for the next k -symbol fractional differential equation:

Δ k , τ ν φ ( τ , ζ ) + ( φ + μ φ 2 ) φ = ϑ ∣ ς ∣ φ 2 .

To end this part, we have to prove ℵ ∈ L ∞ ( [ 0 , ⊤ ] , L ∞ ) . Equation (3.17) leads to the conclusion

1 − ⊤ ν ⁄ k Γ k ( ν + 1 ) ( ‖ ℵ ‖ + μ ‖ ℵ ‖ 2 ) ‖ ℵ ‖ ≤ ϑ φ 0 2 ‖ ς 0 ‖ .

But ∣ ℵ ∣ ≤ 1 ; therefore, we obtain

(3.18) ‖ ℵ ‖ ≤ ϑ φ 0 2 ‖ ς 0 ‖ 1 − ⊤ ν ⁄ k ( 1 + μ ) Γ k ( ν + 1 ) .

Hence, as a consequence, ℵ ∈ L ∞ ( [ 0 , ⊤ ] , L ∞ ) . This holds because ς ∈ L ∞ ( [ 0 , ⊤ ] , L ∞ ) , and consequently, φ ∈ L ∞ ( [ 0 , ⊤ ] , L ∞ ) .

Step 5. Second bound estimates of υ . Differentiate the second evolution equation in System (3.1) with respect to ζ and we obtain

(3.19) γ Δ k , τ ν ∂ ς ∂ ζ = δ ∂ 2 υ ∂ ζ 2 − ∂ ς ∂ ζ φ − ∂ φ ∂ ζ ς = δ ∂ 2 Ψ ∂ ζ 2 − δ β ∂ ς ∂ ζ − ∂ ς ∂ ζ φ − ∂ φ ∂ ζ ς .

In addition, we differentiate the third evolution equation in System (3.1) with respect to ζ and obtain

(3.20) Δ k , τ ν ∂ φ ∂ ζ = ϑ ∂ ∣ ς ∣ ∂ ζ φ 2 + 2 ( ϑ ∣ ς ∣ − 1 ) φ ∂ φ ∂ ζ − 3 μ φ 2 ∂ φ ∂ ζ .

Multiplying equations (3.19) and (3.20) by ∂ ς ∂ ζ and ∂ φ ∂ ζ accordingly, integrating over the set O ¯ , accumulating up and utilizing the fact that φ and ς are in L ∞ ( [ 0 , ⊤ ] , L ∞ ) , we receive that

Δ k , τ ν γ ∂ ς ∂ ζ ( τ , ⋅ ) L 2 2 + ∂ φ ∂ ζ ( τ , ⋅ ) L 2 2 ≤ C ν ⁄ k ∫ O ¯ ∂ 2 Ψ ∂ ζ 2 ∂ ς ∂ ζ + ∂ ς ∂ ζ 2 + ∂ φ ∂ ζ ∂ ς ∂ ζ + ∂ ∣ ς ∣ ∂ ζ ∂ φ ∂ ζ + ∂ φ ∂ ζ 2 ( τ , ⋅ ) .

But ς ∈ L 2 ( [ 0 , ⊤ ] , H 1 ) , then the Young integral inequality yields

(3.21) Δ k , τ ν γ ∂ ς ∂ ζ ( τ , ⋅ ) L 2 2 + ∂ φ ∂ ζ ( τ , ⋅ ) L 2 2 ≤ C ν ⁄ k ∂ ς ∂ ζ ( τ , ⋅ ) L 2 2 + ∂ φ ∂ ζ ( τ , ⋅ ) L 2 2 + ∂ 2 Ψ ∂ ζ 2 ( τ , ⋅ ) L 2 2 .

The facts ς , φ ∈ L ∞ ( [ 0 , ⊤ ] , H 1 ) and ς 0 , φ 0 ∈ H 1 ( O ¯ ) , and by the generalized Gronwall inequality [21] together with equation (3.11), we have that υ ∈ L ∞ ( [ 0 , ⊤ ] , H 1 ) ∩ L 2 ( [ 0 , ⊤ ] , H 2 ) .

Step 6. Approximated outcome. We create a series of approximate solutions for the System (3.1) in this step. Consider the following system:

(3.22a) α Δ k , τ ν υ n ( τ , ζ ) = β ∂ 2 υ n ∂ ζ 2 + ∂ ς n ∂ ζ

(3.22b) γ Δ k , τ ν ς n ( τ , ζ ) = δ ∂ υ n ∂ ζ − φ n − 1 ς n + δ ε

(3.22c) Δ k , τ ν φ n ( τ , ζ ) = ( − 1 + ϑ ∣ ς n ∣ ) φ n − 1 φ n − μ φ n − 1 φ n 2

subjected with the HBC

υ n ( τ , 0 ) = 0 , υ n ( τ , 1 ) = 0 , 0 ≤ τ ≤ ⊤ ,

and the initial condition ( υ n 0 , ς n 0 , φ n 0 ) = ( υ 0 , ς 0 , φ 0 ) .

We proceed to show that System (3.22) has only one solution ( υ n , ς n , φ n ) in the same functional spaces ( C ( [ 0 , ⊤ ] ; H 1 ) ∩ L 2 ( [ 0 , ⊤ ] ; H 2 ) × C ( [ 0 , ⊤ ] ; H 1 ) × C ( [ 0 , ⊤ ] ; H 1 ) ) . To achieve this, we divided System (3.22) into two subsystems: (3.22a)–(3.22b), and (3.22c), respectively.

Assume that ( υ n 1 , ς n 1 ) and ( υ n 2 , ς n 2 ) are a couple solutions taking the property υ n = υ n 1 − υ n 2 and ς n = ς n 1 − ς n 2 and that they satisfy

(3.23a) α Δ k , τ ν υ n ( τ , ζ ) = β ∂ 2 υ n ∂ ζ 2 + ∂ ς n ∂ ζ ,

(3.23b) γ Δ k , τ ν ς n ( τ , ζ ) = δ ∂ υ n ∂ ζ − φ n − 1 ς n .

By multiplying (3.23a) by υ n and integrating over the domain O , we conclude that

(3.24) α ν ⁄ k + 1 Δ k , τ ν ‖ υ n ( τ , ⋅ ) ‖ L 2 2 + β ∂ υ n ∂ ζ ( τ , ⋅ ) L 2 2 = ∫ O ∂ ς n ∂ ζ υ n ( τ , ⋅ ) .

By multiplying (3.23b) by ς n and integrating over the domain O , to obtain

(3.25) γ ν ⁄ k + 1 Δ k , τ ν ‖ ς n ( τ , ⋅ ) ‖ L 2 2 + ‖ φ n − 1 ς n ( τ , ⋅ ) ‖ L 2 2 = δ ∫ O ς n ∂ υ n ∂ ζ ( τ , ⋅ ) ,

Using estimates (3.24) and (3.25) and integration by parts, as well as the observation that υ n exits on the boundary, the result is obtained.

(3.26) Δ k , τ ν [ δ α ‖ υ n ( τ , ⋅ ) ‖ L 2 2 + λ ‖ ς n ( τ , ⋅ ) ‖ L 2 2 ] + ‖ φ n − 1 ς n ( τ , ⋅ ) ‖ L 2 2 + δ β ∂ υ n ∂ ζ ( τ , ⋅ ) L 2 2 = 0 ,

which gives υ n = 0 and ς n = 0 . Thus, System (3.23) admits a unique bound uniform solution over ( C ( [ 0 , ⊤ ] ; H 1 ) ∩ L 2 ( [ 0 , ⊤ ] ; H 2 ) ) × C ( [ 0 , ⊤ ] ; H 1 ) .

We aim to prove that φ n belongs to C ( [ 0 , ⊤ ] ; H 1 ) with ℜ ( φ n ) ≥ 0 . Equation (3.22c) can be written as follows:

(3.27) Δ k , τ ν φ n ( τ , ζ ) = Σ ( τ , φ n , ζ ) ,

φ n ∣ τ = 0 = φ 0 ,

where Σ : [ 0 , ⊤ ] × C × O → C , which is continuous with respect to first and second variables and locally Lipschitz with respect to its third variable. The existence of unique outcome can be realized by the Cauchy-Lipschitz theorem [24] with φ 0 ( ζ ) . Suppose that [ 0 , ⊤ * ) is the maximal interval containing the outcome for τ ≤ ⊤ * . Then we obtain ℜ ( φ n ) ≥ 0 , as in Step 1. In addition, the following inequality

(3.28) ∣ Δ k , τ ν φ n ( τ , ζ ) ∣ ≤ ϑ ∣ ς n ∣ ∣ φ n − 1 ∣ ∣ φ n ∣ ≤ ϑ ‖ ς n ‖ L ∞ ‖ φ n − 1 ‖ L ∞ ∣ φ n ∣ ,

implies that ς n and φ n − 1 are in the space C ( [ 0 , ⊤ ] ; H 1 ) . Consequently, the Gronwall Lemma indicates that φ n bounded on [ 0 , ⊤ * ] . This means that there occurs a unique solution (the maximum solution is unique) in [ 0 , ⊤ * ] . In view of (3.22a) and (3.22b), we obtain

(3.29) Δ k , τ ν [ δ α ‖ υ n ( τ , ⋅ ) ‖ L 2 2 + γ ‖ ς n ( τ , ⋅ ) ‖ L 2 2 ] + ‖ φ n − 1 ς n ( τ , ⋅ ) ‖ L 2 2 + δ β ∂ υ n ∂ ζ ( τ , ⋅ ) L 2 2 = δ ε ς n ^ ( τ ) .

While, equation (3.22c) yields

(3.30) Δ k , τ ν ‖ φ n ( t , ⋅ ) ‖ L 1 + ∫ O ( ∣ φ n − 1 ∣ ∣ φ n ∣ ) ( τ , ⋅ ) + ν 2 ∫ O ( ∣ φ n − 1 ∣ ∣ φ ∣ n 2 ) ( τ , ⋅ ) ≤ ϑ 2 2 ν ‖ φ n − 1 ς n ( τ , ⋅ ) ‖ L 2 2 .

Summing (3.29) and (3.30), this leads to

(3.31) Δ k , τ ν [ δ α ‖ ς n ( τ , ⋅ ) ‖ L 2 2 + γ ‖ ς n ( τ , ⋅ ) ‖ L 2 2 ] + μ ϑ 2 ‖ φ n ( τ , ⋅ ) ‖ L 1 + 1 2 ‖ φ n − 1 ς n ( τ , ⋅ ) ‖ L 2 2 + δ β ∂ υ n ∂ ζ ( τ , ⋅ ) L 2 2 ≤ C ν ⁄ k ε ‖ ς n ( τ , ⋅ ) ‖ L 2 .

That is the solution ( υ n , ς n , φ n ) is bounded. Therefore, we can design the approximated outcome in ( υ n , ς n , φ n ) instead of ( υ , ς , φ ) , and the related auxiliary functions p n and Ψ n . Now, the estimate of the suggested solution is as follows: ( υ n , ς n , φ n )

(3.32) max n max τ ∈ [ 0 , ⊤ ] ( ‖ υ n ( τ , ⋅ ) ‖ L 2 + ‖ ς n ( τ , ⋅ ) ‖ L 2 + ‖ φ n ( τ , ⋅ ) ‖ L 1 ) ≤ C ˜

and

(3.33) max n max τ ∈ [ 0 , ⊤ ] ( ‖ Ψ n ( τ , ⋅ ) ‖ H 1 + ‖ ς n ( τ , ⋅ ) ‖ L ∞ + ‖ φ n ( τ , ⋅ ) ‖ L ∞ ) + ‖ Ψ n ‖ L 2 ≤ C ˜ ,

where C ˜ a positive constant depends on all the parameters of System (3.1).

In the same manner of Step 5, we obtain the following conclusion:

(3.34) Δ k , τ ν γ ∂ ς n ∂ ζ ( τ , ⋅ ) L 2 2 + ∂ φ n ∂ ζ ( τ , ⋅ ) L 2 2 ≤ C ˜ ∂ ς n ∂ ζ ( τ , ⋅ ) L 2 2 + ∂ φ n ∂ ζ ( τ , ⋅ ) L 2 2 + ∂ φ n − 1 ∂ ζ ( τ , ⋅ ) L 2 2 + ∂ 2 Ψ n ∂ ζ 2 L 2 2 .

Putting

ϒ n ( τ ) ≔ ∂ ς n ∂ ζ ( τ , ⋅ ) L 2 2 + ∂ φ n ∂ ζ ( τ , ⋅ ) L 2 2 .

By operating (3.34) by L k ν , we obtain

(3.35) ϒ n ( τ ) ≤ ϒ 0 + C ˜ ∫ 0 τ ( τ − t ) ν ⁄ k − 1 k Γ k ( ν ) ϒ n ( t ) d t + C ˜ ∫ 0 τ ( τ − t ) ν ⁄ k − 1 k Γ k ( ν ) ϒ n − 1 ( t ) d t + C ˜ ‖ Ψ n ‖ L 2 2 ≤ C ˜ ν ⁄ k , 0 + C ˜ ν ⁄ k , 1 ∫ 0 τ ( τ − t ) ν ⁄ k − 1 k Γ k ( ν ) ϒ n − 1 ( t ) d t ≤ L ˜ + L ˜ ∫ 0 τ ( τ − t ) ν ⁄ k − 1 k Γ k ( ν ) ϒ n − 1 ( t ) d t ,

where L ˜ ≔ max ( C ˜ ν ⁄ k , 1 , C ˜ ν ⁄ k , 0 ) . Induction iteration implies that

(3.36) ϒ n ( t ) ≤ L ˜ ∑ j = 0 n − 1 ( L ˜ τ ) j k Γ k ( ( ν ⁄ k ) j + 1 ) + ( L ˜ τ ) n k Γ k ( ( ν ⁄ k ) n + 1 ) .

A computation gives

(3.37) ϒ n ( τ ) ≤ L ˜ Ξ ν ⁄ k , 1 ( L ˜ τ ) ,

where Ξ ν ⁄ k , 1 ( . ) indicates the k -symbol Mittag-Leffler function [25]. It follows that

(3.38) max n max τ ∈ [ 0 , ⊤ ] ϒ n ( τ ) ≤ L ˜ Ξ ν ⁄ k , 1 ( L ˜ ⊤ ν ⁄ k ) .

Hence, from (3.33) and (3.38), we obtain

(3.39) max n max τ ∈ [ 0 , ⊤ ] ( ‖ υ n ( τ , ⋅ ) ‖ H 1 + ‖ ς n ( τ , ⋅ ) ‖ H 1 + ‖ φ n ( τ , ⋅ ) ‖ H 1 ) + ‖ υ n ‖ L 2 ≤ L

and

(3.40) max n ( ‖ Δ k , τ ν υ n ( τ , ⋅ ) ‖ L 2 + ‖ Δ k , τ ν ς n ( τ , ⋅ ) ‖ L 2 + ‖ Δ k , τ ν φ n ( t , ⋅ ) ‖ L 2 ) ≤ L ⊤ ν ⁄ k .

Step 7. Convergence of the roughly solution (approximated). The constraints set forth in the preceding phases, specifically (3.39) and (3.40), mandate that we have weak convergence at least up to the extraction of a subsequence

( υ υ n , ς n , φ n ) → ( υ , ς , φ ) , weakly in L ∞ ( [ 0 , ⊤ ] , H 1 ) 3 .

The next conclusion is to establish a strong convergence in L ∞ ( [ 0 , ⊤ ] , L 2 ( O ¯ ) ) 3 . Let v n * = v n − v n − 1 and prepare the evolution equations for ( υ n * , ς n * , φ n * )

(3.41a) α Δ k , τ ν υ n * ( τ , ζ ) = β ∂ 2 υ n * ∂ ζ 2 + ∂ ς n * ∂ ζ ,

(3.41b) γ Δ k , τ ν ς n * ( τ , ζ ) = δ ∂ υ n * ∂ ζ − φ n − 1 ς n * − φ n − 1 * ς n − 1 ,

(3.41c) Δ k , τ ν φ n * ( τ , ζ ) = ( − 1 + ϑ ∣ ς n − 1 ∣ ) ( φ n − 1 * φ n + φ n − 1 φ n * ) − μ φ n − 1 φ n * ( φ n + φ n − 1 ) − μ φ n − 1 2 φ n − 1 * + ϑ ∣ ς n * ∣ φ n φ n − 1 .

Since ( υ n , ς n , φ n ) fulfills the inclusions (3.2) and (3.3), then similarly for ( υ n * , ς n * , φ n * ) . A multiplication of equations (3.41a), (3.41b), and (3.41c), accordingly by υ n * , ς n * , φ n * , integration over O and summation up introduce the following description:

Δ k , τ ν ( δ α ‖ υ n * ( τ , ⋅ ) ‖ L 2 2 + γ ‖ ς n * ( τ , ⋅ ) ‖ L 2 2 + ‖ φ n * ( τ , . ) ‖ L 2 2 ) ≤ ∫ O Π ( ∣ ς n − 1 ∣ , ∣ ς n ∣ , ∣ φ n − 1 ∣ , ∣ φ n ∣ ) ,

where Π indicates a positive real function. Assume that

P n ( τ ) ≔ ‖ υ n * ( τ , ⋅ ) ‖ L 2 2 + ‖ ς n * ( τ , . ) ‖ L 2 2 + ‖ φ n * ( τ , ⋅ ) ‖ L 2 2 .

In view of equation (3.32) on ∣ ς n − 1 ∣ , ∣ ς n ∣ , ∣ φ n − 1 ∣ , ∣ φ n ∣ , the Young integral inequality leads to

(3.42) Δ k , τ ν P n ( τ ) ≤ C ˜ ( P n ( τ ) + P n − 1 ( τ ) ) .

In virtue of the generalized fractional Gronwall lemma, equation (3.42) becomes

(3.43) P n ( τ ) ≤ T ˜ ν ⁄ k ∫ 0 τ P n − 1 ( s ) d s ,

where the constant T ˜ ν ⁄ k depends on the parameters of system (3.1) and its initial condition. Therefore, this implies that

P n ( t ) ≤ ( T ˜ ν ⁄ k τ ) n − 1 ( n − 1 ) ! sup s ∈ [ 0 , ⊤ ] P 1 ( s ) ,

which yields the sequence ( υ n , ς n , φ n ) is Cauchy in L ∞ ( [ 0 , ⊤ ] , L 2 ( O ¯ ) ) 3 . As a consequence, we obtain that φ n − 1 → φ strongly. Hence, System (3.1) has a solution.

Step 8. Uniqueness. Suppose that ( υ 1 , ς 1 , φ 1 ) and ( υ 2 , ς 2 , φ 2 ) are two solution of System (3.1) under the same initial condition ( υ 0 , ς 0 , φ 0 ) ∈ H 1 ( O ¯ ) . Assume the following functions υ = υ 1 − υ 2 , ς = ς 1 − ς 2 and φ = φ 1 − φ 2 , such that

(3.44a) α Δ k , τ ν υ ( τ , ζ ) = β ∂ 2 υ ∂ ζ 2 + ∂ ς ∂ ζ ,

(3.44b) γ Δ k , τ ν ς ( τ , ζ ) = δ ∂ υ ∂ ζ − φ ς ,

(3.44c) Δ k , τ ν φ ( τ , ζ ) = ( − 1 + ϑ ∣ ς ∣ ) φ 2 − μ φ 3 .

Multiplying equations (3.44a), (3.44b), and (3.44c), accordingly by υ , ς , φ , integrating over O ¯ and summing up, we have

Δ k , τ ν ( δ α ‖ υ ( τ , ⋅ ) ‖ L 2 2 + γ ‖ ς ( τ , ⋅ ) ‖ L 2 2 + ‖ φ ( τ , ⋅ ) ‖ L 2 2 ) ≤ ε ˜ ν ⁄ k ( ‖ ς ‖ L 2 2 + ‖ φ ‖ L 2 2 ) .

The Gronwall lemma gives the uniqueness of solution. This is finished the proof.□

Remark 3.2

When k = 1 , we obtain a result regarding the classical fractional calculus (Riemann-Liouville fractional calculus). Moreover, when k = 1 , the fractional parameter ν = 1 , φ = 0 and ζ is a real variable and we obtain the result in [11].

3.2 Special cases

In this section, we assume that υ ( τ , ζ ) is an approximated function by the rotated Koebe function [26]:

ρ τ ( ζ ) = ζ ( 1 − τ ζ ) 2 , ∣ ζ ∣ < 1 , τ ∈ [ 0 , 1 ] .

Also, we assume that System (3.1) is subjected to the non-HBC υ ( τ , 0 ) = 0 and υ ( τ , 1 ) = ε σ ( τ ) , ε > 0 . In the theory of geometric functions, the extreme convex function in the open unit disk is called the Koebe function (certain varieties of univalent function in the unit disk). The movement of flow in a rotating plate (the unit disk) is described in many fluid systems. As a result, comprehending rotating fluids is crucial for comprehending some geophysical phenomena, such ocean currents and air circulation. A rotating tank experiment can be used to replicate these systems on a smaller scale in a laboratory environment to gain insights into more extensive natural processes.

We have the following result:

Theorem 3.3

Consider ( υ 0 , ς 0 , φ 0 ) ∈ H 1 ( O ¯ ) 3 with ℜ ( φ 0 ) ≥ 0 . If System (3.1) satisfies the inequality

( 1 + μ ) k Γ k ( ν + 1 ) < 1 , ν ∈ ( 0 , 1 ] ,

then it admits a unique global outcome ( υ , ς , φ ) under non-HBC υ ( τ , 0 ) = 0 and υ ( τ , 1 ) = ε σ ( τ ) , where

(3.45) ( υ , ς , φ ) ∈ ( C ( [ 0 , 1 ] ; H 1 ) ∩ L 2 ( [ 0 , 1 ] ; H 2 ) × C ( [ 0 , 1 ] ; H 1 ) × C ( [ 0 , 1 ] ; H 1 ) )

and ℜ ( φ ) ≥ 0 for all ζ ∈ O ¯ and τ ∈ [ 0 , 1 ] . Moreover,

(3.46) ( Δ k , τ ν υ , Δ k , τ ν ς , Δ k , τ ν φ ) ∈ ( L 2 ( [ 0 , 1 ] ; L 2 ) × C ( [ 0 , 1 ] ; L 2 ) × C ( [ 0 , 1 ] ; L 2 ) ) .

Proof

The proof is similar to the proof of Theorem 3.1 for an arbitrary value of ε . □

Remark 3.4

The function υ in Theorem 3.3 is univalent then in view of [27, Theorem 3], there is a sequence of univalent polynomials ϱ m 1 ( ⋅ , ζ ) ∣ υ such that ϱ m 1 ( ⋅ , O ) ∣ υ ⊂ υ ( ⋅ , O ) . Note that the non-HBC in Theorem 3.3 can be written by a convex formula:

( 1 − σ 1 ) υ ( τ , 0 ) + σ 2 υ ( τ , 1 ) = 0 , σ 1 , σ 2 ∈ ( 0 , 1 ) .

Theorem 3.5

If System (3.1) satisfies the inequality

( 1 + μ ) k Γ k ( ν + 1 ) < 1 , ν ∈ ( 0 , 1 ] ,

then it admits a unique univalent global outcome ( υ , ς , φ ) under non-HBC υ ( τ , 0 ) = 0 and υ ( τ , 1 ) = ε σ ( τ ) , where

(3.47) ( υ , ς , φ ) ∈ ( C ( [ 0 , 1 ] ; H 1 ) ∩ L 2 ( [ 0 , 1 ] ; H 2 ) × C ( [ 0 , 1 ] ; H 1 ) × C ( [ 0 , 1 ] ; H 1 ) )

and

(3.48) ( Δ k , τ ν υ , Δ k , τ ν ς , Δ k , τ ν φ ) ∈ ( L 2 ( [ 0 , 1 ] ; L 2 ) × C ( [ 0 , 1 ] ; L 2 ) × C ( [ 0 , 1 ] ; L 2 ) ) .

Furthermore, there are sequences of univalent polynomials such that

( ϱ m 1 ( ⋅ , O ) , ϱ m 2 ( ⋅ , O ) , ϱ m 3 ( ⋅ , O ) ) ⊂ ( υ ( ⋅ , O ) , ς ( ⋅ , O ) , φ ( ⋅ , O ) ) .

3.3 Convergence toward the stable point

The convergence to the steady point of fluid refers to the process by which a fluid system reaches a stable state after experiencing some form of disturbance or change. In this context, a steady state refers to a condition where the fluid properties such as velocity, pressure, and temperature remain constant over time and space.

The convergence to the steady state of fluid can be described by the Navier-Stokes equations, which govern the behavior of fluid flow. These equations describe the conservation of mass, momentum, and energy in the fluid system, and their solutions can be used to predict the behavior of the fluid over time.

The rate at which a fluid system converges to its steady state depends on various factors, such as the initial conditions of the system, the viscosity of the fluid, and the presence of external forces such as gravity or pressure gradients. In some cases, the convergence to the steady state may be rapid, while in others, it may take a long time, or the system may never reach a true steady state at all.

Numerical simulations and experimental observations can be used to study the convergence to the steady state of fluid in different systems, such as fluid flow through pipes, rivers, and oceans. Understanding this process is important in a range of applications, including fluid dynamics, hydrodynamics, and weather forecasting, where the ability to predict the behavior of fluid systems is essential for making accurate predictions and informed decisions.

This step deals with the convergence of the outcome ( υ , ς , φ ) to the origin ( 0 , 0 , 0 ) in H 1 ( O ¯ ) × L ∞ ( O ¯ ) × L ∞ ( O ¯ ) .

Theorem 3.6

If System (3.1) satisfies ε = 0 and ℜ ( φ 0 ) ≠ 0 , then

(3.49) ‖ υ ( τ , ⋅ ) ‖ H 1 + ‖ ς ( τ , ⋅ ) ‖ L ∞ + ‖ φ ( τ , ⋅ ) ‖ L ∞ → 0 .

Proof

Three steps will comprise the proof. The first step derives a lower estimate of fluidity (LEF) φ . Note that fluidity is related to the viscosity of a fluid, which is a scale of its strength to flow. The more viscous a fluid is, the more it resists flow and the less fluid it appears. For example, honey is more viscous than water, which means it flows more slowly and has a thicker consistency. While the second step shows the convergence in L 2 ( O ¯ ) and the third step imposes the convergence in L ∞ ( O ¯ ) .

Step 1. LEF φ . LEF depends on the material and the conditions in which it is being tested. For a material to have a measurable fluidity, it must have some level of viscosity. Therefore, materials with very low viscosity, such as gases, have a high fluidity, while materials with high viscosity, such as solids, have a low fluidity. However, this can vary depending on the specific material and the conditions under, which it is being tested.

Since ℜ ( φ 0 ) ≠ 0 , then by analytically of φ 0 in ( H 1 ) , there is a nonempty closed set O 0 in O ¯ , where φ 0 does not vanish. We rearrange the evolution (3.1a) on φ as follows:

(3.50) Δ k , τ ν φ − 1 ( τ , ζ ) = ( 1 − ϑ ∣ ς ∣ ) φ 2 + μ φ 3 ,

but in view of Step 4 in Theorem 3.1, the L ∞ -limits of φ and ς are uniform for all τ ≤ ⊤ ; therefore, for all ζ ∈ O 0 , we obtain

Δ k , τ ν ∣ φ − 1 ( τ , ζ ) ∣ ≤ c .

Consequently, we obtain

(3.51) ∣ φ ( τ , ζ ) ∣ ≥ k Γ k ( ν + 1 ) k Γ k ( ν + 1 ) ∣ φ 0 ∣ + c ⊤ ν ⁄ k , τ ∈ [ 0 , ⊤ ] .

This implies LEF.

Step 2. Convergence in L 2 ( O ¯ ) . In view of (3.6) and (3.9), we obtain

(3.52) Δ k , τ ν [ δ α ‖ υ ( τ , ⋅ ) ‖ L 2 2 + γ ‖ ς ( τ , ⋅ ) ‖ L 2 2 ] + ‖ φ ς ( τ , ⋅ ) ‖ L 2 2 + δ β ‖ υ ζ ( τ , ⋅ ) ‖ L 2 2 = 0

and

(3.53) Δ k , τ ν [ δ α ‖ υ ( t , ⋅ ) ‖ L 2 2 + γ ‖ ς ( t , ⋅ ) ‖ L 2 2 ] + μ ϑ 2 ‖ φ ( τ , ⋅ ) ‖ L 1 + 1 2 ‖ φ ς ( τ , ⋅ ) ‖ L 2 2 + δ β ‖ υ ζ ( τ , ⋅ ) ‖ L 2 2 = 0 .

Joining (3.52) and (3.53) and employing the Gronwall lemma, we have

(3.54) lim τ → ∞ ( ‖ υ ( τ , ⋅ ) ‖ L 2 2 + ‖ ς ( τ , ⋅ ) ‖ L 2 2 + ‖ φ ( τ , ⋅ ) ‖ L 1 ) ⟶ 0 .

This completes the convergence of solution in L 2 ( O ¯ ) .

Step 3. Convergence in L ∞ ( O ¯ ) . Combining (3.13) and (3.15) and then utilizing the L ∞ -bound of φ lead to

Δ k , τ ν ∂ Ψ ∂ ζ ( τ , ⋅ ) L 2 2 + γ ∣ ς ( τ , ζ ) ∣ 2 + ∂ 2 Ψ ∂ ζ 2 ( τ , ⋅ ) L 2 2 + ∣ ς ( τ , ζ ) ∣ 2 ≤ η ‖ υ ( τ , ⋅ ) ‖ L 2 2 + ‖ ς ( τ , ⋅ ) ‖ L 2 2 + ∂ Ψ ∂ ζ ( τ , ⋅ ) L 2 2 ,

where the constant η > 0 be resultant from all the connections in System (3.1). Applying the Gronwall lemma to obtain

(3.55) lim τ → ∞ ‖ ς ( t , ⋅ ) ‖ L ∞ 2 = 0 .

Putting the convergence (3.55) in equation (3.1c), we obtain

(3.56) lim τ → ∞ ‖ φ ( τ , ⋅ ) ‖ L ∞ 2 = 0 .

As a consequence, we obtain

( υ , ς , φ ) ∈ H 1 ( O ¯ ) × L ∞ ( O ¯ ) × L ∞ ( O ¯ ) .

This finishes the proof.□

Remark 3.7

To ensure that behaviors or solutions are dependable and consistent, convergence toward a stable point is essential in the study and optimization of processes and systems. Convergence toward a stable point is the method used in this article to describe how the system behaves over time. In Step 2, we demonstrate that for tiny enough perturbations, the outcome stays small. In Step 3, we demonstrate that the result converges to the steady state even for minor perturbations and that the measure of convergence is exponential.

4 Conclusion

We demonstrated an analytic technique in this study for proving the existence and uniqueness of k-symbol fractional differential systems in complex domains. The suggested approach is based on the k-symbol fractional Duhamel principle, which is applicable to numerous fractional systems. We had employed the homogeneous boundary value problem all throughout the article. Stability and convergence of solutions are discussed, as well. We have indicated that for a small perturbation, the solution remains in the same domain. Whether little alterations or disruptions to an initial flow will cause it to expand or contract over time is referred to as the stability of solutions. The structure of the Navier-Stokes equations is complicated, and forecasting the patterns of fluid flows requires an understanding of the stability of their solutions. One may consider the nonhomogeneous example in the future study.

Funding information: This work is not funded.
Author contributions: All authors read and agreed to the published version of the manuscript.
Conflict of interest: The authors declare no conflict of interest.
Data availability statement: Data sharing not applicable to this article as no data-sets were generated or analyzed during the current study.

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Received: 2023-03-19

Revised: 2023-11-27

Accepted: 2024-05-17

Published Online: 2025-01-27

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/dema-2024-0029

Keywords for this article

fractional differential equations; fractional differential operator; K-symbol calculus; fractional calculus; analytic functions; the Cauchy-Schwartz inequality

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